Delicate Comparison of the Central and Non-Central Lyapunov Ratios with Applications to the Berry–Esseen Inequality for Compound Poisson Distributions

For each t ∈ ( − 1 , 1 ) , the exact value of the least upper bound H ( t ) = sup { E | X | 3 / E | X − t | 3 } over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment E X 1 / E X 1 2 = t , and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum M : = sup L 1 ( X ) / L 1 ( X − E X ) = 17 + 7 7 / 4 , where L 1 ( X ) = E | X | 3 / ( E X 2 ) 3 / 2 is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M . As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables X 1 , X 2 , … is proven in terms of the central Lyapunov ratio L 1 ( X 1 − E X 1 ) with the constant 0.3031 · H t ( 1 − t 2 ) 3 / 2 ∈ [ 0.3031 , 0.4517 ) , t ∈ [ 0 , 1 ) , which depends on the normalized first-moment t : = E X 1 / E X 1 2 of random summands and being arbitrarily close to 0.3031 for small values of t , an almost 1.5 size improvement from the previously known one.